Locally Bounded Family
A set (also called a family) U of real-valued or complex-valued functions defined on some topological space X is called locally bounded if for any x0 in X there exists a neighborhood A of x0 and a positive number M such that
for all x in A and f in U. In other words, all the functions in the family must be locally bounded, and around each point they need to be bounded by the same constant.
This definition can also be extended to the case when the functions in the family U take values in some metric space, by again replacing the absolute value with the distance function.
Read more about this topic: Local Boundedness
Famous quotes containing the words locally, bounded and/or family:
“To see ourselves as others see us can be eye-opening. To see others as sharing a nature with ourselves is the merest decency. But it is from the far more difficult achievement of seeing ourselves amongst others, as a local example of the forms human life has locally taken, a case among cases, a world among worlds, that the largeness of mind, without which objectivity is self- congratulation and tolerance a sham, comes.”
—Clifford Geertz (b. 1926)
“I could be bounded in a nutshell and count myself a king of
infinite space, were it not that I have bad dreams.”
—William Shakespeare (1564–1616)
“The family is in flux, and signs of trouble are widespread. Expectations remain high. But realities are disturbing.”
—Robert Neelly Bellah (20th century)